An analysis of Wolfenstein parametrization for the Kobayashi-Maskawa matrix shows that it has a serious flaw: it depends on three independent parameters instead of four as it should be. Because this approximation is currently used in phenomenological analyzes from the quark sector, the reliability of almost all phenomenological results is called in question. Such an example is the latest PDG fit from \cite{CA}, p. 150. The parametrization cannot be fixed since even when it is brought to an exact form it has the same flaw and its use lead to many inconsistencies.

The Dita paper claims that the Wolfenstein parameterization is defective because its apparent four real degrees of freedom are redundant; instead there are only three. Such a defect would prevent the parameterization from exploring “almost all” of the space of possible 3×3 unitary matrices. Instead of the whole 4-dimensional real manifold of 3×3 unitary matrices (up to multiplication of rows and columns by complex phases), one would obtain only a 3-dimensional submanifold.

In particular, the paper claims that it is impossible to use the Wolfenstein parameterization to obtain a unitary 3×3 matrix with the magnitude of all amplitudes the same (and equal to sqrt(1/3) ). This is the “democratic unitary 3×3 matrix”, a subject Marni Sheppeard and I have explored at length. It took me a few minutes to verify that it is possible to set these parameters (lambda, A, rho, and eta) to obtain a unitary matrix with all magnitudes equal.
So I wrote up a one page paper giving the solution and sent it up to arXiv: Comment on “On one parametrization of Kobayashi-Maskawa matrix.” Did they accept it? No. And as of 24 hours later, no explanation.

So here’s my view of the situation. ArXivhad no problem whatsoever with publishing a paper that accused hundreds of phenomenology papers of being based on a very basic error. But they refused to publish a paper showing that the standard view of the CKM matrix is correct.

This is not the case of an amateur writing an insane paper claiming that the majority view on physics is wrong. It’s the reverse; an amateur is correcting an obviously defective paper written by a professional. Clearly arXiv cares more about the institution of the authors who contribute papers than they care about the actual content.

I’ve given my first lecture on physics to students at ITT Technical Institute, so I suppose I’m technically again in academia and can no longer claim amateur status. Also, I’ve got the final proofs back from IJMPD on my gravity paper, and I’m now quite convinced it will appear this month, so soon I should have my first published, peer reviewed, paper. This is after spending 25 years in the “real world”.

To read the ones in between, you have to find the links that take you to the next / previous blog post. They’re at the top of the post, just above where it gives the date and time of the post.

If I run out of useful things to do I’ll write these up as a paper and try to publish it as it might be interesting to those who mess around with E8. But my belief is that symmetry should not be at the foundations; it’s only a convenient tool for solving math problems.

If it interests you, I wanted to get your take on a recent paper, “An unconventional space-time model of electrons and its application to the many-electron problem,” by Werner Hofer. Here’s the link:

I like it. It has some things in common with your approach to particles, using geometric algebra.

Hofer mentions some difficulty describing the detection of individual electrons. I think there’s a way around that using shared waves. (As Bohm suggested, I think it’s necessary to treat electrons as collective beasts. Looking at an isolated electron is like listening to the sound of one hand clapping)

I like it. It has some things in common with your approach to particles, using geometric algebra.

Hofer mentions some difficulty describing the detection of individual electrons. I think there’s a way around that using shared waves. (As Bohm suggested, I think it’s necessary to treat electrons as collective beasts. Looking at an isolated electron is like listening to the sound of one hand clapping)

I rather doubt it relates directly to this matrix issue, but I recently encountered a bizarre apparent paradox relating to degrees of freedom, and I’ve been meaning to post a question about it on sci.math.research. (Sadly, sci.math is a bit too much of a bear pit these days to expect useful replies to a question like it.)

Anyway, I was trying to find a general rational parametrization of the pair of equations:

x + y + z + t = 0 (zero)
x.y.z.t = 1

which I had derived by means of birational transforms from a system I am really interested in.

By further such transforms I managed to express it in the birationally equivalent form (reassigning t, i.e. a “new” t):

1 + p^2 = r^2
1 + q^2 = s^2
r.s = t^2

Well, thought I, obviously the next step is to take

r, s, t = P.Q^2, P.R^2, P.Q.R (*)

which satisfies the last generally, and plug these back in the first pair.

But doing that, and dividing throughout by the fourth powers that arise gives two equations each of the form:

X^4 + Y^2 = P^2

(Bear with me – I know this is a bit long for a possibly OT post, but we’re nearly there!)

I then found a general rational parametrization to the latter, with P unconstrained, in other words a solution of the following form, with a parameter P :
X, Y = X(P,U), Y(P,U)

But as P is unconstrained, this solution “kills two birds with one stone” so to speak, and can be applied to the first pair of equations above using a pair of independent parameters, U, V, one for each.

But here’s where is gets weird – The three equations above (or the original pair of equations) have only two degrees of freedom, whereas we have found a general solution with *three* independent parameters!

Not it’s easy to see how this arises – In the equations (*) one of Q and R is disposable, because it can be absorbed in P. But I’m fairly sure that “wiggle room” is lost in the three-parameter solution.

In other words, I don’t think there is any birational transform that can eliminate one of the parameters, and yield a general two-parameter solution one would expect.

I suppose an algebraic geometer might just say that the original system is equivalent to two independent points on a line, because that is what the solution seems to be telling us. But I’m not sure, and it’s rather perplexing all the same.

Apologies if this seems bizarrely OT, and by all means delete if so. I posted it only to share my bewilderment, and because conceivably the same kind of phenomenon (if it exists) crops up in this matrix problem.